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Proof step 67 from proof validrefs-put-notin-01 (previous step)

is_array(str, r1, st), r ∈ st, okval(st[r -- mkfs(class, ty1, fieldvar) '], ty1), r -- mkfs(class, ty1, fieldvar) ' ∈ st, okval(val, ty1), st[r1 -- _type] .type ≤ ty, ∀ r2, fs0, val0, st0, str0, r3, ty0. ¬ r2 ∈ acc ∧ fs0 ' ≠ _type ∧ fs0 ' ≠ _length ∧ okval(val0, fs0 .type) ∧ r2 -- fs0 ' ∈ st0 ∧ okval(st0[r2 -- fs0 '], fs0 .type) → (validrefs(str0, acc + r3 ', ty0, st0[r2 -- fs0 ', val0]) ↔ validrefs(str0, acc + r3 ', ty0, st0)) ⊦ r1 = jvmref, r = r1, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty1, fieldvar) ', val]) ↔ validrefs(str, (r1 ' + acc) + r0 ', ty, st), mkfs(class, ty1, fieldvar) ' = _type, mkfs(class, ty1, fieldvar) ' = _length, r ∈ acc

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 66:
is_array(str, r1, st) ∧ st[r1 -- _type] .type ≤ ty, is_array(str, r1, st), r ∈ st, okval(st[r -- mkfs(class, ty1, fieldvar) '], ty1), r -- mkfs(class, ty1, fieldvar) ' ∈ st, okval(val, ty1), st[r1 -- _type] .type ≤ ty, ∀ r2, fs0, val0, st0, str0, r3, ty0. ¬ r2 ∈ acc ∧ fs0 ' ≠ _type ∧ fs0 ' ≠ _length ∧ okval(val0, fs0 .type) ∧ r2 -- fs0 ' ∈ st0 ∧ okval(st0[r2 -- fs0 '], fs0 .type) → (validrefs(str0, acc + r3 ', ty0, st0[r2 -- fs0 ', val0]) ↔ validrefs(str0, acc + r3 ', ty0, st0)) ⊦ r1 = jvmref, r = r1, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty1, fieldvar) ', val]) ↔ acc + r0 ' = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r1 -- _length] .intval ∧ is_referencevalue(st[r1 -- i ']) ∧ (acc + r0 ') .first = st[r1 -- i '] .refval ∧ validrefs(str, acc + r0 ', st[r1 -- _type] .type .type, st)), mkfs(class, ty1, fieldvar) ' = _type, mkfs(class, ty1, fieldvar) ' = _length, r ∈ acc

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st, ty, acc + r0 ', r1, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st, ty, acc + r0 ', r1, str)]]